I would want:
-Continue FPS style
-Use stats more like Fallout 1&2 than the 3 method
-More weapons; it's okay to have more than 1-2 guns that use the same ammo type
-More accurate weapons: I like being able to hit things past 10 yards. To compensate, give enemies longer range perception. Oh yeah, and laser rifles need to not suck so much.
-More character interaction
-More exploration element; make vaults bigger and more expansive, potentially have sub-plots based around derelict vaults/ruins (examples: Vault 15 and Sierra Army Depot from Fallout 2)
-Setting: There are a lot of options here. It might be interesting to see it take place in a downtown environment, with block after block of huge skyscrapers (hopefully with less blocked streets this time, and more expansive above-ground building exploration). Most major cities could do this.
-Radioactivity: amount depends a bit on timeline, but there should be less emphasis on it than in 3. Essentially, only a few big areas and much of the food would give radiation, and ideally the player wouldn't start with a Geiger counter to know about it (so it should take longer to accrue, but be harder to detect; make player actually pay attention to Radiation symbols).
-Inventory: Actually, I'm a little divided on this one. While I kinda appreciate the limited ammo storage when it has weight, the weightless ammo system was also damned convenient. Either one works, I suppose, as long as they keep it close to any of the existing Fallout inventory systems.
Parallel Streaks said:
And last of all, please don't make the BoS into the generic good guys again. In the first two they weren't particularly nice, they were just trying to find new technology, and were JUST as elitist as the Enclave.
I could agree with this. The Brotherhood of Steel was always the "Good Guys" but they've also had some moral ambiguity. Even as late as Tactics, many members came off as bloodthirsty, racist, elitist, and uncaring to the plight of those living in the wastes. Frankly, I find that more appealing and easier to relate to than the cartoony "Good Guy" feel they had in Fallout 3.
Edit: I also agree with Grampy_bone up there, more information displays would be nice; all the previous Fallout games had a full list of stats for every weapon as well as a brief (occasionally sarcastic) written description of every item. It'd be nice to see more of that charm and humor that was lacking in #3.